Bounded cohomology of transformation groups

Let M be a complete connected Riemannian manifold of finite volume. We present a new method of constructing classes in bounded cohomology of transformation groups such as Homeo0(M,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Homeo}_0(M,\mu )$$\end{document}, Diff0(M,vol)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Diff}_0(M,\hbox {vol})$$\end{document} and Symp0(M,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Symp}_0(M,\omega )$$\end{document}. As an application we show that for many manifolds (in particular for hyperbolic surfaces) the third bounded cohomology of these groups is infinite dimensional.